CHAPTER 15

Proportions on the GMAT

A baseball team won 45 percent of the first 80 games it played. How many of the remaining 82 games will the team have to win in order to have won exactly 50 percent of all the games it played?

Above is a typical Problem Solving proportions question. In this chapter, we’ll look at how to apply the Kaplan Method to this question, discuss the proportions rules being tested, and go over the basic principles and strategies that you want to keep in mind on every Quantitative question involving proportions. But before you move on, take a minute to think about what you see in this question and answer some questions about how you think it works:

PREVIEWING PROPORTIONS ON THE GMAT

Proportions show up on the GMAT in the form of fractions, ratios, decimals, and percents. Fundamentally, proportions represent relationships. You deal with proportions somehow every day, even if you might not realize it. Rates and speeds, for example, are expressed as proportions: a car travels 30 miles per hour, or a worker earns $80 per task. You can use these proportional relationships to determine how many miles a car travels at this speed over a given length of time, or how many tasks a worker must complete in order to earn a given amount of money. Prices are also proportions: a certain item costs $4.50, so how much would 24 of the same item cost? But proportions can exist between any values: the ratio of gerbils to parrots in a pet store might be 2:3; in other words, you may not be given the exact number of gerbils or parrots, but you do know that for every 2 gerbils the store has, there are 3 parrots as well.

Often the GMAT tests proportions in the form of word problems, so the translation skills you learned in the Algebra chapter (a form of Paraphrasing) will continue to be valuable here as you cut through the seeming novelty of the given “story” to understand the underlying proportional relationships being described.

What Are the Various Ways You Can Express 45 Percent (as a Decimal, Fraction, or Ratio)?

One of the most important skills you can develop for dealing with proportions questions on the GMAT is the ability to convert quickly among percent, decimal, fractional, and ratio forms of the same value. This skill is so crucial because GMAT problems often mix these various formats within the same problem—getting all the values expressed in the same way will make them easier to handle. Also, certain values are just easier to work with in one format or another. For instance, you’d much rather perform arithmetic operations with the fraction image than you would with the unwieldy decimal 0.1428571 …

In this problem, you may choose to express 45 percent as a decimal (0.45); as a fraction (image, which reduces to image); or as a ratio (45:100, which reduces to 9:20), depending on which is easiest to work with at the time. Remembering that percent simply means “out of 100” is the key to making these conversions. Later in this chapter you will find a chart to help you memorize some of the common conversions you’re most likely to need on Test Day.

What Are Two Different Ways to Solve This Question? Which One Is Most Efficient?

A lot of information is given in this question, so it is especially important that you organize your scratchwork effectively. You are asked for the number of games a team still needs to win in order to achieve a certain winning percentage. You are given the team’s winning percentage so far, the number of games it’s played so far, and the number of games it still needs to play, so you have all the information you need to solve arithmetically.

As is usual on the GMAT, this question doesn’t necessarily give you the information in the most convenient form, so you will have to do some calculating to figure out the actual number of games the team has won so far and the total number of games the team will have played, but these calculations are possible, so you can solve this question arithmetically. (We will return to this question and solve it using the Kaplan Method later this chapter.)

If that approach was the one that first occurred to you, that’s fine—and you’re not alone. You have likely been rewarded all your academic life for knowing how to do the “schoolroom math,” and this approach can get you the correct answer on the GMAT as well. What might not have occurred to you initially is the other strategy you could use to solve this problem: Backsolving. Whenever you are asked to solve for a single value (here, it’s the number of games the team still needs to win) and the answer choices are all simple numerical values, it may be quicker and easier to answer the question by plugging the values in the answer choices into the scenario described in the question stem.

More guidance on Backsolving is available in the Problem Solving chapter of this book; it’s a highly efficient strategy, and it’s well worth becoming comfortable with, even if you are generally expert at solving using “classroom math.”

What Trap Answer Choices Might the Testmakers Include for Those Who Misread the Question?

There are various unknowns in this question that the unwary test taker might mistakenly choose as the correct answer—but to the wrong question. As you practice, never choose an answer choice simply because you recognize that number from your calculations; some of these choices are likely to be traps.

This question asks you for the number of additional games the team must win. Possible trap answers are the number of wins so far—choice (A)—and the total number of wins the team needs—choice (E).

The final step of the Kaplan Method for Problem Solving is to confirm your answer; by double-checking that you are answering the question that was asked, you will avoid this all-too-common mistake.

What GMAT Core Competencies Are Most Essential to Success on This Question?

As you’ve seen throughout the Quant section, Critical Thinking is necessary for deciding on the most efficient strategic approach, and Attention to the Right Detail will ensure that you answer the question that is asked instead of falling for trap answers. Specifically on proportions questions, you will use your skill in Paraphrasing to convert ratios into the form that’s easiest for that problem. You will also pay Attention to the Right Detail so that you don’t confuse different types of ratios such as part-to-part and part-to-whole.

Here are the main topics we’ll cover in this chapter:

Handling GMAT Proportions

Now let’s apply the Kaplan Method to the proportions question you saw earlier:

A baseball team won 45 percent of the first 80 games it played. How many of the remaining 82 games will the team have to win in order to have won exactly 50 percent of all the games it played?

Step 1: Analyze the Question

You are given the percentage of games a team has won for the first 80 games and the number of games it has yet to play (82). You are asked to calculate how many more games the team needs to win to have a 50% win record.

Step 2: State the Task

You need to calculate the total number of games that will be played, then the total number of those that need to be won for a 50% record. Then calculate how many games have already been won and subtract that from the total number of wins needed.

Step 3: Approach Strategically

First, add the 80 games that have already been played to the number of remaining games to get the total games: 80 + 82 = 162. To win 50% of the games, the team would need to win half of these, or 81. You can use the percentage given to calculate how many games the team has already won: 45% of 80.

The number of the games the team still needs to win is 81 – 36 = 45. Choice (B) is correct. 45% can be expressed as image = image. Use this fraction when multiplying: image × 80 = 9 × 4 = 36.

The answer choices are small, manageable numbers, so you may want to use Backsolving, especially if you’re not sure how to set up the arithmetic. If you choose to Backsolve, remember to think about exactly what the answer choices represent: the number of additional games the team needs to win. Start with choice (B), 45 games. Calculate the games already won, 45% of 80 = image × 80 = 36, and add the 45 additional games: 36 + 45 = 81.

Now you can ask yourself, is 81 half (or 50%) of all the games? The team’s already played 80, and there are 82 left to play, so 80 + 82 = 162 and, yes, choice (B) is 50% of 162.

If you had instead Backsolved starting with (D), you would have ended up with too high a proportion of games won (91 of 162) and known that you need a smaller number of wins during the second part of the season in order to end up with a winning percentage of 50%.

Step 4: Confirm Your Answer

Review your calculations to be sure they’re correct. You can also verify that your answer is logical using Critical Thinking: the team would need more than 40 additional wins to make up for the fact that it won less than 50% of its first 80 games.

Now let’s look at each of the areas of proportions that show up on the GMAT Quantitative section, starting with applying fractions to proportions.

APPLYING FRACTIONS TO PROPORTIONS

The GMAT will frequently ask you to deal with proportional relationships by working with fractions used to represent ratios. We will deal more specifically with how ratios are tested on the GMAT later in this chapter, but in the meantime, let’s look at how to apply the skills you learned in the Arithmetic chapter for dealing with fractions to questions that ask about proportions.

A proportion is a comparison of two ratios. Usually, a proportion consists of an equation in which two ratios (expressed as fractions) are set equal to each other. You can set up proportions whenever you are given a relationship between more than one fraction or ratio and are asked to solve for the missing part. Fractions in proportions represent part-to-part or part-to-whole relationships. When dealing with proportions, pay close attention to what something is a proportion of. A common “twist” is to use different denominators in the same problem.

In the proportion image=image, a and d are identified as the extremes, and c and b are identified as the means. These terms stem from the alternate notation for these ratios, a:b = c:d, in which the extremes occupy the end positions of the equation. But despite that terminology, it’s usually easier to express proportions as fractions when solving GMAT problems.

To solve a proportion, use cross multiplication to multiply the numerator from the first ratio by the denominator from the second ratio, then the denominator from the first ratio by the numerator of the second ratio, and set the values equal to each other. In other words, set the product of the means equal to the product of the extremes. After this step, divide both sides of this new equation to isolate the variable you’re solving for.

Example:  Solve for m:image=image.

Using cross multiplication, the equation becomes 6 × 33 = m × 11. Before you spend time multiplying out 6 × 33, notice that you can instead divide both sides of the equation by 11. This simplifies to 6 × 3 = m. Thus, m = 18.

In word problems, set up the proportions so that units of the same type are either one above the other or directly across from one another.

Example:  The ratio of T-shirts to sweaters in a closet is 4:5. If there are 12 T-shirts in the closet, how many sweaters are there?

Set up a proportion to solve this question, being careful to line up the units. There are different ways you can approach this problem to get the correct answer. One possible proportion would be image. Note that the label of T-shirts appears in both numerators and the label of sweaters appears in both denominators. Another possible proportion would be image, where the numbers corresponding with T-shirts appear on the left side and the values corresponding with sweaters appear on the right side. Either way, cross multiplying the means and the extremes results in the equation 4x = 60. Therefore, x = 15. There are 15 sweaters in the closet.

In-Format Question: Applying Fractions to Proportions on the GMAT

Now let’s use the Kaplan Method on a Data Sufficiency question that involves applying fractions to proportions:

After 360 liters of fuel were added to a container, the amount of fuel in the container was image of the tank’s capacity. What is the capacity of the container?

(1)  After the 360 liters were added, the container had 108 liters less than image of the tank’s capacity.

(2)  Before the 360 liters were added, there were 180 liters of fuel in the container.

Step 1: Analyze the Question Stem

First, recognize that this is a Value question. You need an exact value of the capacity of the container after 360 liters of fuel are added to it. Next, take control of the question by setting up an equation. Call the capacity of the container C liters and say that before the 360 liters of fuel were added, there were N liters of fuel in the container. Since after the 360 liters of fuel were added, the amount of fuel in the container was image of the tank’s capacity, write the equation N + 360 = imageC. Now look at the statements.

Step 2: Evaluate the Statements Using 12TEN

Start with Statement (2), which looks like the more straightforward of the two. Statement (2) says that before the 360 liters were added, there were 180 liters of fuel in the container. After the 360 liters were added, there were 180 + 360 = 540 liters. Since after the 360 liters were added, the amount of fuel in the container was image of the tank’s capacity, you can write the equation 540 = image C. This is a first-degree equation, so it must lead to exactly one value for C. Don’t waste time calculating to find the exact value. All you need to know is that Statement (2) is sufficient. Eliminate (A), (C), and (E).

Now look at Statement (1). Statement (1) says that after the 360 liters were added, the container had 108 liters less than image of the tank’s capacity. You can translate this as N + 360 = imageC − 108. Since you know from the question stem that N + 360 = imageC, you can say that imageC = imageC − 108. This is a first-degree equation, so it must lead to exactly one value for C. Statement (1) is sufficient. Eliminate choice (B).

If you want to check your work, you can solve the equation imageC=imageC−108 for C to be sure that there is just one solution. Multiplying both sides by 8, you have image−8(108), 5C = 6C – 864, and 864 = C. You see that there is indeed just one solution. This statement is sufficient.

Either statement is sufficient, making (D) the correct answer.

Notice that both statements lead to the same capacity of 864 liters for the container. The statements in a Data Sufficiency question will never contradict one another, so if each statement alone is sufficient, the two statements will always answer the question in the same exact way. You can use this fact to double-check whether you are applying the statements correctly.

TAKEAWAYS: APPLYING FRACTIONS TO PROPORTIONS

 

Practice Set: Applying Fractions to Proportions on the GMAT

Answers and explanations at end of chapter

  1. The cost of Brand V paper is proportional to the weight. If 18 ounces of Brand V paper cost $3.06, what is the cost of 24 ounces of Brand V paper?

    • $4.08

    • $5.58

    • $6.12

    • $7.31

    • $7.59

  2. The volume of a certain substance is always directly proportional to its weight. If 48 cubic inches of the substance weigh 112 ounces, what is the volume, in cubic inches, of 63 ounces of this substance?

    • 27

    • 36

    • 42

    • 64

    • 147

  3. To fill an art exhibit, the students in an art course are assigned to create one piece of artwork each in the following distribution: image are sculptures, image are oil paintings, image are watercolors, and the remaining 10 pieces are mosaics. How many students are in the art class?

    • 80

    • 120

    • 240

    • 320

    • 400

RATIOS

A ratio is a comparison of two quantities by division. Ratios may be written with a fraction bar image, with a colon (x:y), or in English (“the ratio of x to y”). Ratios can (and in most cases, should) be reduced to lowest terms, just as fractions are reduced.

Example:  Joe is 16 years old and Mary is 12. The ratio of Joe’s age to Mary’s age is 16 to 12.

image

In a ratio of two numbers, the numerator is often associated with the word of and the denominator with the word to.

image

Example:  In a box of doughnuts, 12 are sugar and 18 are chocolate. What is the ratio of sugar doughnuts to chocolate doughnuts?

image

Part-to-Part Ratios and Part-to-Whole Ratios

A ratio represents the proportional relationship between quantities or numbers. A ratio can compare either a part to another part or a part to a whole. One type of ratio can readily be converted to the other only if all the parts together equal the whole and there is no overlap among the parts (that is, if the whole is equal to the sum of its parts).

Example:  The ratio of domestic sales to foreign sales of a certain product is 3:5. What fraction of the total sales are the domestic sales?

First note that this is the same as asking for the ratio of the amount of domestic sales to the amount of total sales.

In this case, the whole (total sales) is equal to the sum of the parts (domestic and foreign sales). You can convert from a part:part ratio to a part:whole ratio. For every 8 sales of the product, 3 are domestic and 5 are foreign. The ratio of domestic sales to total sales is then image or 3:8.

Example:  The ratio of domestic to European sales of a certain product is 3:5. What is the ratio of domestic sales to total sales?

Here you cannot convert from a part:part ratio (domestic sales:European sales) to a part:whole ratio (domestic sales:total sales) because you don’t know whether there are any other sales besides domestic and European sales. The question doesn’t say that the product is sold only domestically and in Europe, so you cannot assume there are no other sales. The ratio asked for here cannot be determined.

Ratios with More Than Two Terms

Some GMAT questions deal with ratios that have three or more terms. Always represent them with colons (e.g., x:y:z).

Even though they convey information about more relationships, ratios involving more than two terms are governed by the same principles as two-term ratios. Ratios involving more than two terms are usually ratios of various parts, and it is usually the case that the sum of these parts equals the whole, making it possible to find part-to-whole ratios as well.

Example:  Given that the ratio of men to women to children in a room is 4:3:2, what other ratios can be determined?

The whole here is the number of people in the room, and since every person is either a man, a woman, or a child, you can determine part:whole ratios for each of these parts. Of every nine (4 + 3 + 2 = 9) people in the room, 4 are men, 3 are women, and 2 are children. This gives you three part:whole ratios:

Ratio of men:total people = 4:9 or image.

Ratio of women:total people = 3:9 = 1:3 or image.

Ratio of children:total people = 2:9 or image.

In addition, from any ratio of more than two terms, you can determine various two-term ratios among the parts.

Ratio of women:men = 3:4 or image.

Ratio of men:children = 4:2 = 2:1 or image.

Ratio of children:women = 2:3 or image.

And finally, if you were asked to establish a relationship between the number of adults and the number of children in the room, this would also be possible. For every 2 children, there are 4 men and 3 women, which is 4 + 3 = 7 adults. Thus,

Ratio of children:adults = 2:7 or image.

Ratio of adults:children = 7:2 or image.

Naturally, a GMAT question will require you to determine only one or at most two of these ratios, but knowing how much information is contained within a given ratio will help you to determine quickly which questions are solvable and which, if any, are not.

Ratios versus Actual Numbers

Ratios are always reduced to simplest form. If a team’s ratio of wins to losses is 5:3, this does not necessarily mean that the team has won exactly 5 games and lost exactly 3. For instance, if a team has won 30 games and lost 18, its ratio is still 5:3. Unless you know the actual number of games played (or the actual number won or lost), you can’t know the actual values of the parts in the ratio.

Example:  In a classroom of 30 students, the ratio of the boys in the class to students in the class is 2:5. How many students are boys?

You are given a part-to-whole ratio (boys:students). This ratio can also be expressed as a fraction. Multiplying this fraction by the actual whole gives the value of the corresponding part. There are 30 students, and image of them are boys, so the number of boys must be image×30. (Note: it’s usually most helpful to set up ratio scratchwork in columns so you see the relationships clearly.)

image

You can make some deductions about the actual values, though, even when you aren’t given a total directly. Often, this is the key to solving a challenging proportions question.

Let’s say that a car dealership has used and new cars in stock in a ratio of 2:5. You don’t know the actual number. You know that if there are 2 used cars, there will be 5 new cars. And if there are 4 (or 2 × 2) used cars, there will be 10 (or 5 × 2) new cars. If there are 6 used cars (2 × 3), then there are 15 new cars (5 × 3). In other words, the actual numbers will always equal the figures in the ratio multiplied by the same (unknown) factor.

You can translate from ratios into algebra:

used:new = 2:5            

          used = 2x; new = 5x

Example:  A homecoming party at College Y is initially attended by students and alumni in a ratio of 1 to 5. But after two hours, 36 more students arrive, changing the ratio of students to alumni to 1 to 2. If the number of alumni did not change over those two hours, how many people were at the party when it began?

Besides the 36 additional students who arrive, nowhere are actual numbers referred to directly. But you can jot down the two ratios you’re given in a slightly more useable form:

image

Translate into algebra. Let x be the number of students initially at the party.

image

Now you’ve got some equations to work with:

A = 5x

       A = 2x + 72

You now have two different expressions for A that are equal. Thus,

        5x = 2x + 72

3x = 72

x = 24

Now you know that there were 24 students initially.

Substitute 24 for x to solve for alumni:

    A = 5 × 24

A = 120

The party started with 24 students and 120 alumni, for a total of 144 people.

Guessing with Ratios

You may have noticed that many GMAT ratio problems involve things like students, women, children, cars, and so forth—things that cannot really exist as non-integer values. (You can’t have 43 people at a party, for instance!)

There’s a valuable deduction to be made here—if the entities in a problem cannot logically be non-integers, then the factor by which the ratio is multiplied must be an integer. Therefore, the actual number of something must be a multiple of its value in the ratio.

Example:  

A movie buff owns movies on DVD and on Blu-ray in a ratio of 7:2. If she buys 6 more Blu-ray movies, that ratio would change to 11:4. If she owns movies on no other medium, what was the original number of movies in her library before the extra purchase?

The question asks for the whole of the ratio for which you are initially given only two parts. The question specifies that there are no other parts to the whole, so you can add the two parts together:

DVD:Blu-ray:Total

7:2:9

There’s no such thing as “a fraction of a disc,” so the common factor by which you’d multiply 7, 2, and 9 to find the actual numbers must be an integer. That means the number of DVDs is a multiple of 7, the number of Blu-rays is a multiple of 2, and the total—what you’re looking for—must be a multiple of 9.

Take a look at the answers—only (D) is a multiple of 9, so there’s no need to do any more math; (D) must be the answer.

In-Format Question: Ratios on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with ratios:

The ratio of two quantities is 2:3. If each of the quantities is increased by 6, what is the ratio of the new quantities?

Step 1: Analyze the Question

For ratio questions, keep careful track of which values are ratios and which are actual quantities. You are told that the ratio of quantities is 2:3. So the smallest value for quantity A is 2 and for quantity B is 3. You could also have any multiple of this ratio, so you could have quantity A = 4 and quantity B = 6, for example.

Step 2: State the Task

Now that you have a clear picture of the information, pay Attention to the Right Detail. You are given the ratio of the two quantities, not the exact number of each. Without the actual total value or one of the quantities’ values, you can’t determine the new values. The correct answer is (E).

If you weren’t able to apply this rule so directly, you could solve this question by Picking Numbers for the possible quantities.

Step 3: Approach Strategically

First test the smallest possible quantities, since these will be the most manageable: quantity A = 2 and quantity B = 3. Adding 6 to each quantity, you get the following new values: quantity A = 8 and quantity B = 9. The new ratio would be 8:9.

Before you select (C), however, you must test a second case. If quantity A = 4 and quantity B = 6, then after adding 6 to each quantity, you get the following new values: quantity A = 10 and quantity B = 12. The new ratio would be 10:12, or 5:6.

You can get two distinct results depending on the original values you choose for the actual quantities, so the correct answer must be (E).

Step 4: Confirm Your Answer

In ratio questions, if you change the actual quantities through multiplication or division, the resulting ratio will be the same, regardless of the starting values of the quantities. However, the same ratio may not be preserved if you add or subtract from the quantities. Knowing this rule is a great way to confirm your answer.

TAKEAWAYS: RATIOS

 

Practice Set: Ratios on the GMAT

Answers and explanations at end of chapter

  1. Three investors, A, B, and C, divide the profits from a business enterprise in the ratio of 5:7:8, respectively. If investor A earned $3,500, how much money did investors B and C earn in total?

    • $4,000

    • $4,900

    • $5,600

    • $9,500

    • $10,500

  2. At a given time, what was the ratio of the number of sailboats to the number of motorboats on Lake X?

    • (1)  If the number of motorboats on Lake X had been 25% greater, the number of sailboats on Lake X would have been 110% of the number of motorboats on Lake X.

    • (2)  The positive difference between the number of motorboats on Lake X and the number of sailboats on Lake X was 30.

  3. If the ratio of a to b is 4 to 3 and the ratio of b to c is 1 to 5, what is the ratio of a to c?

    • image

    • image

    • image

    • image

    • image

PERCENTS WITH SPECIFIED VALUES

In the Problem Solving chapter of this book, you learned how to use the Picking Numbers strategy to deal with questions that involve taking percents of unspecified values. Questions that specify the exact values are often more straightforward, but they require your familiarity with several formulas for dealing with fractions and percents.

Key to these questions is the fact that percent is just another word for “per one hundred.”

Therefore, 19% (or 19 percent) means 19 hundredths

or image

or 0.19

or 19 out of every 100 things

or 19 parts out of a whole of 100 parts.

Converting Percents, Fractions, and Decimals

GMAT questions will often require that you convert among percents, decimals, fractions, and ratios. Below you will learn the process for making each of these conversions, but they all boil down to using the percent formula:

image

Plug in the information you have and solve for the missing pieces.

To make a percent from a decimal or fraction, multiply by 100%. Since 100% means 100 hundredths, or 1, multiplying by 100% will not change the number.

Example:  0.17 = 0.17 × 100% = 17%

Example:  image=image×100%=25%

To drop a percent, divide by 100%. Once again, dividing by 100%, which is equivalent to 1, will not change the number.

Example:  32%=image=image=image

Example:  image%=image=image

To change a percent to a decimal, just drop the percent sign and move the decimal point two places to the left. (This is the same as dividing by 100%.)

Example:  0.8% = 0.008.

Being familiar with the following fraction-to-percent equivalents can save you a lot of time on questions that require you to perform such conversions.

Common Fraction-to-Percent Equivalents

image

image

Percent Problems

Percent problems will usually give you two of the terms from the formula below and ask for the third. It is usually easiest to change the percent to a common fraction before performing the calculation. Most percent problems can be solved by plugging into a variant of the percent formula:

Percent × Whole = Part

In percent problems, the whole generally will be associated with the word of, and the part will be associated with the word is. The percent can be represented as the ratio of the part to the whole, or the is to the of.

Example:  What is 25% of 36?

Here you are given the percent and the whole. To find the part, change the percent to a fraction, then multiply. Use the following formula:

image

Since 25% = image, you are really being asked what one-fourth of 36 is.

image× 36 = 9

Example:13 is 33image% of what number?

Recall that 33image%=image

                           13=imagen

                           39 = n

Example:  18 is what percent of 3?

m × 3 = 18

m = 6

6 × 100% = 600%

Other problems will ask you to calculate “percent greater than” or “percent less than.” In these cases, it is the number that follows the word than to which the percentage is applied.

You may find it faster to convert percent greater/less than to percent of. “x% greater than” becomes “(100 + x)% of.” “x% less than” becomes “(100 – x)% of.”

Example:  What number is 15% less than 60?

n = 60 – (15% × 60)

n = 60 – image

n = 60 – 9

n = 51

Example:  What is the price of a television that costs 28% more than a $50 radio?

Let t equal the price of the television and r equal the price of the radio.

t= r + (28% × r)

t= 50 + image

t= 50 + image

t = 50 + 14

t = $64

Percent Increase and Decrease

image

image

New whole = Original whole ± Amount of change

When dealing with percent increase and percent decrease, always be careful to put the amount of increase or decrease over the original whole, not over the new whole.

Example:  If a dress is offered for a discounted price of $120, what is the percent discount if the regular price is $150?

image

You’re asked to calculate the discount, so you’ll use the regular selling price of $150 as the original whole. The difference in prices is $150 – $120, or $30.

image

image

Combining Percents: On some problems, you’ll need to find more than one percent, or a percent of a percent. Be careful: you can’t just add percents, unless the percents are of the same whole. Let’s look at an example.

Example:  The price of an antique is reduced by 20 percent, and then this discount price is further reduced by 10 percent. If the antique originally cost $200, what is its final price?

First, you know that the price is reduced by 20%. That’s the same thing as saying that the price becomes (100% – 20%), or 80% of what it originally was. Eighty percent of $200 is equal to image × $200 or $160. Then, this price is reduced by 10%. So 10% × $160 = $16, and the final price of the antique is $160 – $16 = $144.

A common error in this kind of problem is to assume that the final price is simply a 30 percent reduction of the original price. That would mean that the final price is 70 percent of the original, or 70% × $200 = $140. But, as you’ve just seen, this is not correct. Adding or subtracting percents directly only works if those percents are being taken of the same whole. In this example, since you took 20% of the original price, and then 10% of the reduced price, you can’t just add the percents together.

Note that in this example, a 20% reduction followed by a 10% reduction does not equal a 30% reduction but rather one of 28%. This will always happen—two decreases cause a total percent decrease of less than the sum of the two individual percents. This is because the second percent decrease is taken from a smaller starting value. Likewise, two increases cause a total increase of more than the sum of the two individually, since the second percent increase is calculated based on a higher starting point. You can use this fact to estimate an answer quickly.

In-Format Question: Percents with Specified Values on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question that involves percents with specified values:

A pet store regularly sells pet food at a discount of 10 percent to 30 percent from the manufacturer’s suggested retail price. If during a sale, the store discounts an additional 20 percent from the discount price, what would be the lowest possible price of a container of pet food that had a manufacturer’s suggested retail price of $20.00?

Step 1: Analyze the Question

This question is asking for the lowest possible price of pet food that has been marked down twice. Another way to phrase the question is that you want the maximum possible discount. So you’ll take the maximum discount at each stage of the markdown to determine the lowest price.

Since the question gives you an actual starting value for the price, you cannot Pick Numbers to make the calculations easier. Notice that you could Backsolve this question, since the answer choices are all numbers, but since this question involves multiple discounts, Backsolving could result in lots of calculation. The more efficient approach here is to use what you know about percent decrease.

Step 2: State the Task

Using the starting value of $20, you must apply the two maximum discounts, one after the other.

Step 3: Approach Strategically

The original price for the pet food was $20, and the maximum reduction the store gives is 30%. You can approach the reduction calculation in two ways:

1.   You could determine 30% of $20 and then subtract this amount from $20:

$20 × 0.3 = $6; $20 – $6 = $14

2.   You can find the new price directly. Since 100% – 30% = 70%, the new price is 70% of $20:

$20 × 0.7 = $14

Regardless of the approach, you find that the first reduced price is $14.

Now, you need to take the 20% discount off this reduced price. Again, you could apply either approach to calculate this value, but the second approach is more efficient: $14 × 0.8 = $11.20. (B) is correct.

Step 4: Confirm Your Answer

Since there wasn’t much arithmetic, you can quickly check for any careless errors. Also, reread the question stem to make sure that you didn’t misinterpret anything and that you solved for the correct value.

TAKEAWAYS: PERCENTS WITH SPECIFIED VALUES

 

Practice Set: Percents with Specified Values on the GMAT

Answers and explanations at end of chapter

  1. Each employee of Company X is a member of precisely 1 of 3 shifts of employees. Among the 60 members of the first shift, 20 percent participate in the pension program; among the 50 members of the second shift, 40 percent participate in the pension program; and among the 40 members of the third shift, 10 percent participate in the pension program. What percent of the workers at Company X participate in the pension program?

    • 20%

    • 24%

    • 36%

    • 37.5%

    • 70%

  2. In 2010, a basketball team won 30 percent of its 20 basketball games. In 2011, the team won 28 percent of its 25 basketball games. What was the percent increase from 2010 to 2011 in the number of basketball games the team won?

    • 5%

    • 14image%

    • 16image%

    • 23image%

    • 60%

    image

  3. In the graph above, A, B, and C represent the sales of major soft drink companies, D represents the sales of store brands, and E represents other soft drink sales. How many dollars’ worth of store-brand soft drinks were sold in region X in 2009?

    • (1)  image=image

    • (2)  In region X, the soft drink sales of major companies totaled $350,000 in 2009.

MIXTURES

Problems involving mixtures may appear intimidating, with their intimations of chemistry, but really they are just proportion, ratio, and percentage problems in disguise. Most mixture questions will ask you to combine two portions that are themselves subdivided into portions. At other times, the question stem may prescribe the ratio of elements in a mixture and then add or remove some or all of one of those portions, asking you to recalculate the new ratio that results. Recognize the familiar patterns at work and you will be a step ahead on the GMAT.

Mixtures on the GMAT are most commonly mixtures of liquids.

Example:  You currently have 10 oz. of a solution that is 14 percent alcohol and the rest water. How much water would you have to add to the solution to make the alcohol content equal to 10 percent?

If the solution is 14% alcohol now, that means there are 10 oz. × image = 1.4 oz. of alcohol in the initial solution. The rest of the solution is water.

Think critically about this situation, since it’s one that occurs often in mixtures problems: When water is added, the amount of alcohol in the mixture remains constant at 1.4 oz. However, you are told that this same amount of alcohol represents 10% of the resulting mixture.

Since 1.4 is 10% of 14, the final mixture must have a total volume of 14 oz. The difference in volume between the old and new mixtures is accounted for solely by the addition of water, so you’d need to add 14 – 10 = 4 oz. of water to make a 10% alcohol solution.

Notice that the only real difference between this and a lot of the proportion or ratio questions that you’ve seen so far in this chapter is that it deals with a liquid, rather than with cars, students, adults, or children. Mixture problems involve many of the same skills you use on other ratio problems, except they require you to keep track of amounts rather than numbers.

In-Format Question: Mixtures on the GMAT

Now let’s use the Kaplan Method on a Problem Solving question dealing with mixtures:

Two brands of detergent are to be combined. Detergent X contains 20 percent bleach and 80 percent soap, while Detergent Y contains 45 percent bleach and 55 percent soap. If the combined mixture is to be 35 percent bleach, what percent of the final mixture should be Detergent X?

Step 1: Analyze the Question

This is a complex question, but there is a straightforward solution. You are creating a new mixture from two others, X and Y. X is 20% bleach, and Y is 45% bleach. The new mixture is to be 35% bleach.

In other words, some amount of a 20% bleach mixture plus some amount of a 45% bleach mixture will balance each other out to form a 35% bleach mixture.

Step 2: State the Task

Because this question involves finding a particular balance between Detergents X and Y, you can use the balanced average approach to solve. You could also use Algebra or Backsolving, but the balanced average approach will be the most efficient. This will let you calculate the proportion of Detergent X in the final mixture.

Step 3: Approach Strategically

The question does not state how many parts of Detergent X are used, so call this x. And the question does not state how many parts of Y are used, so call this y. Here’s how it balances:

image

Since the amount above the average has to equal the amount below the average, 0.10y = 0.15x. To solve for a proportional amount, you can view this as a ratio. Divide both sides by y and by 0.15 to get the ratio of x to y:

0.10y = 0.15x

image=image

image=image

image=image

So x:y is 2:3. Add the total to the ratio to determine how x relates to the total: x:y:total = 2:3:5.

Thus x:total = 2:5. That means x makes up image, or 40%, of the total mixture. Choice (D) is correct.

Step 4: Confirm Your Answer

Review your calculations for this question. Also confirm that your answer choice makes sense in the context of the question.

TAKEAWAYS: MIXTURES

 

Practice Set: Mixtures on the GMAT

Answers and explanations at end of chapter

  1. How many liters of water must be evaporated from 50 liters of a 3 percent sugar solution to get a 5 percent sugar solution?

    • 2

    • 4

    • 6

    • 10

    • 20

  2. If the ratio of integers a, b, and c is 1:3:5, what is the value of a + c?

    • (1)  c – a = 12

    • (1)  b – a = 6

  3. A child has a certain number of marbles in a bag: some red, some green, and the rest blue. The number of red marbles in the bag is what percent of the number of green and blue marbles?

    • (1)  The ratio of red to green marbles is 2:3, and the ratio of red to blue marbles is 4:5.

    • (2)  The total number of marbles is 60.

Answer Key

Practice Set: Applying Fractions to Proportions on the GMAT

  1. A

  2. A

  3. C

Practice Set: Ratios on the GMAT

  1. E

  2. A

  3. A

Practice Set: Percents with Specified Values on the GMAT

  1. B

  2. C

  3. D

Practice Set: Mixtures on the GMAT

  1. E

  2. D

  3. A

Answers and Explanations

Practice Set: Applying Fractions to Proportions on the GMAT

1.    (A)

The cost of Brand V paper is proportional to the weight. If 18 ounces of Brand V paper cost $3.06, what is the cost of 24 ounces of Brand V paper?

Step 1: Analyze the Question

We know two things—the cost of Brand V paper is proportional to its weight, and 18 ounces of this paper costs $3.06.

Step 2: State the Task

Calculate the cost of 24 ounces of Brand V paper.

Step 3: Approach Strategically

Let x = the cost of 24 ounces of Brand V paper. The paper cost is proportional to its weight, and we are given three parts of a proportion, so set up the proportion and solve:

image

image

The numerator on the right is now 4 times the numerator on the left, so x must be 4 times the simplified denominator, or $1.02 × 4 = $4.08. Choice (A) is correct.

Step 4: Confirm Your Answer

The answer choices are fairly spread out, so estimation is a great way to confirm our answer. Because 24 is less than one-and-a-half times 18, we need an answer that is also less than one-and-a-half times $3.06. Only (A) comes close.

2.    (A)

The volume of a certain substance is always directly proportional to its weight. If 48 cubic inches of the substance weigh 112 ounces, what is the volume, in cubic inches, of 63 ounces of this substance?

Step 1: Analyze the Question

The language in the first sentence is asking us to translate the question stem into algebra. We can set up an equation relating the volume of an object to its weight.

Step 2: State the Task

Set up the equations from the question stem and solve for the volume of the substance.

Step 3: Approach Strategically

Let’s say that the volume is v cubic inches. Then 48 cubic inches is to 112 ounces as v cubic inches is to 63 ounces. Then image=image. Simplifying the left side, we have this:

image

image

image

image

To solve for x, we can cross multiply:

3 × 63 = 7 × v

3 × 9 = v    

   v = 27

Choice (A) is correct.

Step 4: Confirm Your Answer

You can plug your calculated value for v back into the original equation from the question stem to confirm that your answer makes sense. You can also use estimation to check your work: 63 is slightly more than half of 112, so we need a value for v that is slightly more than half of 48. Among the answer choices, only 27 is a possibility.

3.    (C)

To fill an art exhibit, the students in an art course are assigned to create one piece of artwork each in the following distribution: image are sculptures, image are oil paintings, image are watercolors, and the remaining 10 pieces are mosaics. How many students are in the art class?

Step 1: Analyze the Question

In this question, we are presented with a series of parts that make up the whole—in this case, the number of students in an art class. Notice that most of the whole is identified as fractions of the whole, while one part is identified as a specific quantity. We can use this to our advantage.

Step 2: State the Task

Determine the sum of the fractions in the question stem (sculptures, oil paintings, watercolors), since this makes up all but one part of the whole number of students. Subtracting this fraction from 1 will provide the fraction of the whole that is the remaining part (mosaics). Finally, we will solve for the number of students in the class.

Note that this question can also be solved by Backsolving, since all of the answer choices are numbers and we can test out the answer choices to see if 10 pieces remain after calculating the number of other pieces of artwork.

Step 3: Approach Strategically

We are told that image of the students create sculptures, image create oil paintings, image create watercolors, and the remaining 10 pieces are mosaics. Let’s assume that the total number of students in the class is x. Then the sum of the sculptures, oil paintings, and watercolors is as follows:

image

So the sum of the sculptures, oil paintings, and watercolors is image of the art in the exhibit. Therefore, the remaining 10 pieces must be 1−image=image of the total pieces. We can set up the equation:

imagex = 10; x = 240

Choice (C) is correct.

Step 4: Confirm Your Answer

Plug your value for x into the original equation to confirm your calculations are correct.

Practice Set: Ratios on the GMAT

4.    (E)

Three investors, A, B, and C, divide the profits from a business enterprise in the ratio of 5:7:8, respectively. If investor A earned $3,500, how much money did investors B and C earn in total?

Step 1: Analyze the Question

Three investors, A, B, and C, divide profits in the ratio of 5:7:8, and investor A receives $3,500 in profits.

Step 2: State the Task

We must find the amount earned by investors B and C combined.

Step 3: Approach Strategically

The easiest way to visualize this problem is to consider the profits in terms of equal shares. Investors A, B, and C divide profits in the ratio of 5:7:8, so there are a total of 5 + 7 + 8 = 20 “shares” of profits. Investor A has 5 of these shares, which leaves investors B and C with 20 – 5 = 15 profit shares. Because 15 is 3 times 5, investors B and C combined must have earned three times what investor A earned, or $3,500 × 3 = $10,500. Choice (E) is correct.

Step 4: Confirm Your Answer

When dealing with ratios, be sure to match the given part to the correct part of the ratio.

5.    (A)

At a given time, what was the ratio of the number of sailboats to the number of motorboats on Lake X?

Step 1: Analyze the Question Stem

This Value question stem gives us little information. However, if we let the variable s represent sailboats and m represent motorboats, we can translate the question to image=? Always be careful when translating ratios from English into algebra that the quantities are in the correct position within the fraction.

Step 2: Evaluate the Statements Using 12TEN

Translating Statement (1) into an equation, we get s = (1.25m)(1.10). Because we have a simple equation in which we can isolate image on one side, we could solve for image. This statement is sufficient. Remember to stop your calculations as soon as you have determined sufficiency. We never need to know the actual value of image. We can eliminate (B), (C), and (E).

Translating Statement (2) into an equation, we get |m – s| = 30. We can use Picking Numbers to test the equation.

If we pick m = 31 and s = 1, we get a ratio of 1:31.

If we pick m = 32 and s = 2, we get a ratio of 1:16.

Because we have two possible outcomes, this statement is insufficient. Choice (A) is correct.

6.    (A)

If the ratio of a to b is 4 to 3 and the ratio of b to c is 1 to 5, what is the ratio of a to c?

Step 1: Analyze the Question

We are given two ratios, a to b and b to c, and we are asked to solve for the ratio of a to c.

Step 2: State the Task

We must produce the ratio a to c. We can do so by Picking Numbers or by translating the given information into equations and combining them.

Step 3: Approach Strategically

Translating the first equation, we have image.

Translating the second equation, we have image.

We can calculate image by multiplying image.

Choice (A) is correct.

Had we used Picking Numbers, the most straightforward values to choose are a = 4 and b = 3. If b = 3, then the only permissible value of c is 15. That makes the ratio of a to c equal to image.

Step 4: Confirm Your Answer

When you are given two ratios, their product or quotient will always produce a third ratio. Be sure to confirm the order of the words in the question stem ratios to ensure that you have translated the ratios correctly.

Practice Set: Percents with Specified Values on the GMAT

7.    (B)

Each employee of Company X is a member of precisely 1 of 3 shifts of employees. Among the 60 members of the first shift, 20 percent participate in the pension program; among the 50 members of the second shift, 40 percent participate in the pension program; and among the 40 members of the third shift, 10 percent participate in the pension program. What percent of the workers at Company X participate in the pension program?

Step 1: Analyze the Question

This is a complex percents problem with multiple pieces. We’ll need to take things one at a time to avoid getting overwhelmed.

Step 2: State the Task

We are asked to find the percentage of Company X workers who participate in the pension program.

Step 3: Approach Strategically

There are three shifts of workers, and a different percentage of each participates in the pension program, so we’ll need to calculate each piece separately before combining them to get our answer.

First shift: 20% of the 60 workers participate, or 60 × 0.2 = 12 first-shift workers participate. Second shift: 40% of the 50 workers participate, or 50 × 0.4 = 20 second-shift workers participate. Third shift: 10% of the 40 workers participate, or 40 × 0.1 = 4 third-shift workers participate. That’s a total of 12 + 20 + 4 = 36 workers who participate in the pension program out of the overall 60 + 50 + 40 = 150 workers at Company X.

Now we can plug these totals into the ratio the question asks for: image=image=image=24%. Choice (B) is correct.

Step 4: Confirm Your Answer

If you are pressed for time, you can use estimation on this question. Notice that the two shifts with the most employees have 20% and 40%, respectively, participating in the pension program. Of the two, the 20% is from the larger group, so a rough estimate of these two groups alone puts the answer south of 30% (which would be the midpoint of 20% and 40%). Coupled with the fact that the remaining shift only has a paltry 10% participation rate, we have enough to immediately eliminate (C), (D), and (E).

8.    (C)

In 2010, a basketball team won 30 percent of its 20 basketball games. In 2011, the team won 28 percent of its 25 basketball games. What was the percent increase from 2010 to 2011 in the number of basketball games the team won?

Step 1: Analyze the Question

This question stem contains a lot of information, which we must translate carefully to solve the question. We are asked to calculate the percent increase in the number of games won by the basketball team from 2010 to 2011. To determine this value, we need the actual number of games won in 2010 and won in 2011.

Now that we know what information we need, the first two sentences make sense. We can calculate the number of games won in each year from those two statements.

Step 2: State the Task

Translate the question stem into three equations to determine (1) number of games won in 2010, (2) number of games won in 2011, and (3) percent increase in the number of games won.

Step 3: Approach Strategically

Translating the first sentence (2010), we have 30% × 20 games = 0.30 × 20 games = 6 games.

Translating the second sentence (2011), we have 28% × 25 games = 0.28 × 25 games = 7 games.

The percent increase formula is very important to memorize for Test Day. It is

image

image

(C) is correct.

Step 4: Confirm Your Answer

There are a number of calculations in this solution, so be sure to double-check your work, especially in calculating the percent increase.

9.    (D)

image

In the graph above, A, B, and C represent the sales of major soft drink companies, D represents the sales of store brands, and E represents other soft drink sales. How many dollars’ worth of store-brand soft drinks were sold in region X in 2009?

Step 1: Analyze the Question Stem

This is a Value question. When presented with a chart, take a quick look at the information presented and think about the types of calculations that can be made using the information. The pie chart for this question represents the sales of soft drinks in a given year and region. Notice that two of the percent wedges in the chart are unknown variables n% and p% and that we are given the total sales, $500,000. So we can determine the exact number of sales for three of the five types of drink sales. Also, we know that n and p make up the remaining 24% of the sales or, in equation form, n + p = 24. This is one linear equation for the variables n and p. If we are given another equation, we can determine the value of each variable.

Next, let’s look at the question stem. We are asked to determine the dollar amount of sales D, which is also p% of the total sales. All we need to calculate this amount is the value of p, so we can rephrase the question stem as “What is the value of p?”

Step 2: Evaluate the Statements Using 12TEN

Statement (1) tells us that image=image. Cross multiplying, we have 2p = n. This equation, together with the equation n + p = 24, gives us two different equations with the two unknowns n and p. Because we know that we can solve these equations for p, this statement is sufficient. We can eliminate (B), (C), and (E).

Statement (2) tells us that sales for A, B, and C totaled $350,000. So we can write the equation (34% + 20% + n%)($500,000) = $350,000. We now have an equation with the single variable n. We can solve this equation for the value of n. Then, when we have the value of n, we can substitute this value of n into the equation n + p = 24 to find the value of p. Statement (2) is sufficient. (D) is correct.

Notice that we do not need to calculate the actual value for p or n. Be very aware of this on Test Day so that you use your time most efficiently.

Practice Set: Mixtures on the GMAT

10.   (E)

How many liters of water must be evaporated from 50 liters of a 3 percent sugar solution to get a 5 percent sugar solution?

Step 1: Analyze the Question

We have 50 liters of a 3% sugar solution, and the question tells us that some amount of water is evaporated from the solution.

Step 2: State the Task

We must find the number of liters of water that must evaporate to get a 5% sugar solution.

Step 3: Approach Strategically

We can calculate that 50 liters of a 3% sugar solution has 50 × 0.03 = 1.5 liters of sugar. Since the only thing that changes from the old to the new solution is the amount of water, we know that the amount of sugar remains constant at 1.5 liters. A 5% sugar solution with the same 1.5 liters of sugar would have image liters of water. We must therefore evaporate 50 – 30 = 20 liters of water. Choice (E) is correct.

Step 4: Confirm Your Answer

The answer choices are spread widely apart, so estimation is a great way to confirm our answer. 3% to 5% is almost double the concentration of sugar, so we’ll need to lose almost half of the water. Only (E) comes close.

11.   (D)

If the ratio of integers a, b, and c is 1:3:5, what is the value of a + c?

Step 1: Analyze the Question Stem

For any ratio question, keep careful track of the quantities provided. In this Value question, we are given a ratio of three variables, a, b, and c, and asked to find the sum of a + c. To answer this question, we need to determine the actual quantities of a and c.

Step 2: Evaluate the Statements Using 12TEN

Statement (1) tells us that c – a = 12. From the question stem, we know that the ratio of a to c is 1:5. This means that if a = 1, then c = 5. However, in that case, c – a = 4. Because the numbers are small, we can try the next few multiples of the ratio until we find the pair of numbers with the correct difference.

If a = 2, c = 10; c – a = 8.

If a = 3, c = 15; c – a = 12.

This is the only set of values for this ratio that will have this difference. Statement (1) is sufficient. Eliminate (B), (C), and (E).

Statement (2) tells us that b – a = 6. Using the same logic as in Statement (1), we can find the pair of values for a and b that must be true for the difference to be 6. If we have the actual values for a and b, we can apply the ratio to these values to find the actual value for c. Statement (2) is sufficient. (D) is correct.

12.   (A)

A child has a certain number of marbles in a bag: some red, some green, and the rest blue. The number of red marbles in the bag is what percent of the number of green and blue marbles?

Step 1: Analyze the Question Stem

For this Value question, our first step must be to translate the question. Let r equal the number of red marbles, g equal the number of green, and b equal the number of blue. We want to know what percentage of the green and blue marbles is the number of red marbles, or the value of image.

Step 2: Evaluate the Statements Using 12TEN

We can analyze Statement (1) using Picking Numbers. Suppose there are 4 red marbles. Red:Blue = 4:5, so there are 5 blue marbles. To find the number of green marbles, use the ratio

Red:Green = 2:3

image

image

Green = 6

If there are 4 red, 5 blue, and 6 green marbles, then the question is “4 is what percent of 11?” or “What is image as a percent?” Remember that since this is a Data Sufficiency question, we don’t actually have to do these calculations.

But we should test a second set of numbers. Suppose there are 8 red marbles. Similar reasoning yields 10 blue marbles and 12 green marbles. So now the question is “8 is what percent of 22?” or “What is image as a percent?” Because image reduces to image, the answer is the same as before. We have reason to think the value will be the same for any set of numbers. In that case, Statement (1) is sufficient. Eliminate (B), (C), and (E).

Now let’s evaluate Statement (2). With no relationships among the numbers of different colored marbles, we can pick almost any numbers we want. If there are 1 red marble, 1 green marble, and 58 blue marbles, then the proportion of red marbles to the others is image. But if there are 50 red marbles, 5 blue marbles, and 5 green marbles, then the proportion is image or image. Clearly, different answers are possible. So Statement (2) is insufficient. Choice (A) is correct.

We could also have evaluated both statements using algebra. Algebraically, we can translate the two phrases in Statement (1) as follows:

image

Now substitute for g and b into the expression we are solving for:

image

We now have an expression with a single variable, and that variable will cancel out of both the numerator and denominator. Therefore, we have enough information to answer the question, and Statement (1) is sufficient. Eliminate (B), (C), and (E).

Statement (2) tells us that r + g + b = 60. Therefore, g + b = 60 – r. If we substitute for g + b into the expression we are solving for, then image. In this case, we cannot cancel out r. Therefore, Statement (2) is insufficient. The correct answer is (A).

GMAT BY THE NUMBERS: PROPORTIONS

Now that you’ve learned how to approach proportions questions on the GMAT, let’s add one more dimension to your understanding of how they work.

Take a moment to try the following question. The next page features performance data from thousands of people who have studied with Kaplan over the decades. Through analyzing this data, we will show you how to approach questions like this one most effectively and how to avoid similarly tempting wrong answer choice types on Test Day.

The price of a certain car this year is $42,000, which is 25 percent greater than the cost of the car last year. What was the price of the car last year?

Explanation

The testmakers often set up percentage problems to reward those who do two important things and to punish those who don’t. The first is to be careful about what number the percentage is applied to. The second is to remember that you aren’t a human calculator, and thus look for ways to simplify calculations.

QUESTION STATISTICS
1% of test takers choose (A)
1%of test takers choose (B)
19% of test takers choose (C)
74% of test takers choose (D)
5% of test takers choose (E)
Sample size = 3,799

This problem tells you that an increase of 25% raises the price of a car to $42,000, and then asks for the original price. Be careful to apply that 25% increase to the correct price—the original, not the $42,000. Decreasing $42,000 by 25% yields $31,500, which is the most common wrong answer. Avoid that one error, and your odds of getting the right answer go up dramatically.

If you set up the arithmetic the way the GMAT immediately presents it, you get:

Original price × 1.25 = 42,000

This would have you dividing 42,000 by 1.25; a simple task for a calculator, but an onerous one for a person. You’ll often have more success by converting percentages to fractions. An increase of 25% means an increase of image, bringing the total up to image of the original:

Original price × image = 42,000

Original price = 42,000 × image

Original price = 8,400 × 4

Original price = 33,600

Choice (D) is correct.

More GMAT by the Numbers …

To see more questions with answer choice statistics, be sure to review the full-length CATs in your online resources.